Journal
JOURNAL OF STATISTICAL PHYSICS
Volume 125, Issue 5-6, Pages 1197-1225Publisher
SPRINGER
DOI: 10.1007/s10955-006-9210-z
Keywords
canonical ensemble in quantum theory; probability measures on Hilbert space; Gaussian measures; density matrices
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For a quantum system, a density matrix rho that is not pure can arise, via averaging, from a distribution mu of its wave function, a normalized vector belonging to its Hilbert space H. While rho itself does not determine a unique mu, additional facts, such as that the system has come to thermal equilibrium, might. It is thus not unreasonable to ask, which mu, if any, corresponds to a given thermodynamic ensemble? To answer this question we construct, for any given density matrix rho, a natural measure on the unit sphere in H, denoted GAP(rho). We do this using a suitable projection of the Gaussian measure on H with covariance rho. We establish some nice properties of GAP(rho) and show that this measure arises naturally when considering macroscopic systems. In particular, we argue that it is the most appropriate choice for systems in thermal equilibrium, described by the canonical ensemble density matrix rho(beta)=(1/Z) exp (-beta H). GAP(rho) may also be relevant to quantum chaos and to the stochastic evolution of open quantum systems, where distributions on H are often used.
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